Bounded Entire Functions Theorem Identification

The statement "Every bounded entire function is constant" is a fundamental result in complex analysis.

Understanding Key Terms

• Entire Function: A function that is analytic (holomorphic) over the entire complex plane ($\mathbb{C}$).
• Bounded Function: A function $f(z)$ is bounded if there exists a real number $M > 0$ such that $|f(z)| \le M$ for all $z$ in its domain.

Identifying the Theorem

The theorem directly linking these properties (boundedness and being entire) to the conclusion that the function must be constant is known as Liouville's theorem.

Theorem Statement: Liouville's Theorem

Liouville's theorem states that if a function $f(z)$ is both entire and bounded, then $f(z)$ must be a constant function.

Analysis of Options

• Morera's theorem: This theorem provides a condition for a continuous function to be analytic, based on the integral around closed paths being zero. It is the converse of Cauchy's integral theorem.
• Liouville's theorem: This is the correct theorem, stating that a bounded entire function is constant.
• Cauchy's theorem: This theorem states that the integral of an analytic function over a simple closed curve in a simply connected domain is zero.
• Cauchy's integral formula: This formula relates the value of an analytic function at a point inside a closed curve to the integral of the function over the curve.

Therefore, the correct identification is Liouville's theorem.